Starting from an anisotropic (in all directions including the time direction of the brane) non-SUSY D2 brane solution of type IIA string theory we construct an anisotropic space-like D2 brane (or SD2 brane, for short) solution by the standard trick of a double Wick rotation. This solution is characterized by five independent parameters. We show that compactification on six-dimensional hyperbolic space (H\(_6\)) of a time-dependent volume of this SD2 brane solution leads to accelerating cosmologies (for some time \(t\sim \,t_0\), with \(t_0\) some characteristic time) where both the expansions and the accelerations are different in three spatial directions of the resultant four-dimensional universe. On the other hand at early times (\(t \ll t_0\)) this four-dimensional space, in certain situations, leads to four-dimensional Kasner-like cosmology, with two additional scalars, namely, the dilaton and a volume scalar of H\(_6\). Unlike in the standard four-dimensional Kasner cosmology here all three Kasner exponents could be positive definite, leading to expansions in all three directions.

Eur. Phys. J. C
Anisotropic SD2 brane: accelerating cosmology and Kasner-like space-time from compactification
Kuntal Nayek 0 1
Shibaji Roy 0 1
0 Homi Bhabha National Institute, Training School Complex , Anushakti Nagar, Mumbai 400085 , India
1 Saha Institute of Nuclear Physics , 1/AF, Bidhannagar, Calcutta 700064 , India
Starting from an anisotropic (in all directions including the time direction of the brane) non-SUSY D2 brane solution of type IIA string theory we construct an anisotropic space-like D2 brane (or SD2 brane, for short) solution by the standard trick of a double Wick rotation. This solution is characterized by five independent parameters. We show that compactification on six-dimensional hyperbolic space (H6) of a time-dependent volume of this SD2 brane solution leads to accelerating cosmologies (for some time t ∼ t0, with t0 some characteristic time) where both the expansions and the accelerations are different in three spatial directions of the resultant four-dimensional universe. On the other hand at early times (t t0) this four-dimensional space, in certain situations, leads to four-dimensional Kasnerlike cosmology, with two additional scalars, namely, the dilaton and a volume scalar of H6. Unlike in the standard fourdimensional Kasner cosmology here all three Kasner exponents could be positive definite, leading to expansions in all three directions.
1 Introduction
It is well known [
1
] that a cosmological solution of a
higherdimensional vacuum Einstein equation can give rise to
interesting four-dimensional cosmology (with a period of
accelerated expansion) upon time-dependent hyperbolic space
compactifications [
2–4
]. This process, therefore, evades a no-go
theorem [
5,6
] of obtaining such accelerated expansion in
standard time-independent compactifications. Similar
cosmologies also follow if one includes fluxes [7] and/or a
dilaton field [
8
] in the higher-dimensional theories such as
M/string theory. M/string theory solution which gives rise
to four-dimensional accelerating cosmologies upon
timedependent hyperbolic space compactifications is called the
space-like M2 (SM2) brane (for M theory) or space-like D2
(SD2) brane (for string theory). Space-like branes [
9,10
] are
topological defects localized on a space-like hypersurface
and exist for a moment in time. So, they are time-dependent
solutions of field theories or M/string theory with an isometry
ISO( p+1) × SO(d − p−2, 1) for an S p brane in d space-time
dimensions [
11–14
]. The original motivation for
constructing these solutions was to understand the time-dependent
processes in field and M/string theory [
9,10,15,16
] and also
to have a better understanding of the dS/CFT
correspondence [
17,18
]. The cosmological implication leading to
fourdimensional accelerated expansion from these solutions has
been elucidated in Refs. [
1,7,8,19–24
].
The previous S2 brane solutions considered in the
literature [
7,8,11–14
] were isotropic in the brane directions and
so the four-dimensional accelerating cosmologies obtained
from these solutions were isotropic. In this paper we will
construct an anisotropic SD2 brane solutions of type IIA
string theory and try to see whether similar four-dimensional
accelerating cosmologies can be obtained in all three spatial
directions upon compactification. Another motivation to look
at the anisotropic SD2 brane solution is to see whether one
can get a four-dimensional Kasner-like [25] solution from
it upon compactification where one can get expansions in
all three spatial directions which is not possible in
conventional Kasner solution from four-dimensional vacuum
Einstein equation. The construction of anisotropic SD2 brane
solution follows from the standard double Wick rotation [
26
]
of the known anisotropic non-SUSY D2 brane solution [
27
]
of type IIA string theory. This solution is characterized by
five independent parameters. We then cast the solution in
a suitable time-like coordinate and is given in terms of a
single harmonic function containing a characteristic time t0.
Next, we compactify the space-time on a six-dimensional
hyperbolic space with time-dependent volume. The resultant
metric when expressed in an Einstein frame gives us a
fourdimensional FLRW type space-time with three different scale
factors in three spatial directions. We find that when t ∼ t0,
we can get accelerating cosmologies in all three directions
when other parameters of the solution take some specific
values. Although the expansions and the accelerations in all
three directions are not the same but they do not differ
drastically and the accelerations are all transient. However, when
t t0, the resultant four-dimensional metric takes a
Kasnerlike form when the parameters characterizing the solution
satisfy certain conditions. But because of the presence of the
dilaton as well as the volume scalar of the six-dimensional
hyperbolic space, all the Kasner exponents could be positive
definite, leading to expansions in all three spatial directions.
However, the expansions in this case are decelerating. This
can be contrasted with the standard four-dimensional Kasner
space-time [
25
] (obtained from the solution of the vacuum
Einstein equation) where expansions in all three directions
are not possible.
This paper is organized as follows. In the next section we
give the construction of anisotropic SD2 brane solution from
its time-like counterpart and cast the solution in a
coordinate system suitable for our purpose. In Sect. 3, we obtain
the anisotropic accelerating cosmologies from this solution
upon compactification on six-dimensional hyperbolic space
of a time-dependent volume. In Sect. 4, we show how a
fourdimensional Kasner-like geometry arises from this string
theory solution, where all the Kasner exponents could be
positive definite leading to expansions in all three spatial
directions unlike the standard Kasner solution in four dimensions.
Finally, we conclude in Sect. 5.
2 Anisotropic SD2 brane solutions
In this section we construct the anisotropic SD2 brane
solution from the known anisotropic non-SUSY D2 brane
solution of type IIA string theory. In [
27
], we have constructed an
anisotropic non-SUSY D p brane solution and showed how it
nicely interpolates between a black D p brane and a Kaluza–
Klein “bubble of nothing” when some of the parameters of the
solution are varied continuously and interpreted this
interpolation as closed string tachyon condensation. Here we make
use of that solution and write the anisotropic non-SUSY D2
brane solution in the following by putting p = 2 in Eq. (4)
(we have replaced δ0 by δ3 for convenience) of Ref. [
27
],
3 2
ds2 = F (r ) 8 (H (r )H˜ (r )) 5
×(dr 2 + r 2d 26)
⎧
+F (r )− 85 ⎨ −
⎩
H (r ) δ41 + δ120 + δ130
H˜ (r )
H (r ) δ41 + δ22 + δ23
H˜ (r )
dt 2
+
+
H˜ (r )
H˜ (r )
(4)
(5)
e2(φ−φ0) = F (r ) 21
The metric in the above is given in the Einstein frame. The
various functions appearing in the solution are defined as
F (r ) =
H (r ) α
H˜ (r )
cosh2 θ −
H˜ (r )
H (r )
ω5
H (r ) = 1 + r 5 ,
ω5
H˜ (r ) = 1 − r 5 .
Note that the solution has eight parameters α, β, δ1, δ2, δ3,
θ , ω, and Qˆ . φ0 is the asymptotic value of the dilaton, F[
6
]
is a six form and Qˆ is the magnetic charge associated with
the D2 brane. The solution becomes isotropic in the brane
directions when δ1 = −2δ2 = −2δ3. So, in that sense these
parameters can be called anisotropy parameters. Now for the
consistency of the field equations the eight parameters of the
solution must satisfy the following relations [
27
]:
3
α − β = − 2 δ1
1 2 1 3
2 δ1 + 2 α α + 2 δ1
2 6 2 2
+ 5 δ2δ3 = 5 (1 − δ2 − δ3 )
Qˆ = 5ω5(α + β) sinh 2θ .
These three relations reduce the number of independent
parameters from eight to five, which are ω, θ , and the
anisotropy parameters δ1, δ2 and δ3. Using the second and
the first relations in (4), we can express α and β in terms of
the other parameters as
3 1
α = − 4 δ1 ± 2
3 1
β = 4 δ1 ± 2
458 (1 − δ22 − δ32) − 47 δ12 − 156 δ2δ3
458 (1 − δ22 − δ32) − 47 δ12 − 156 δ2δ3.
The form of the harmonic function H˜ (r ) in (3) indicates that
there is a naked singularity of the solution at r = ω and
therefore, the solution is well defined only for r > ω. Now
we apply the double Wick rotation [
26
] r → i τ , t → −i x 3 to
the solution (1) along with ω → i ω, θ → i θ and θ1 → i θ1,
where θ1 is one of the angular coordinates of the sphere 6
of the transverse space. This operation gives us anisotropic
space-like D2 brane from the anisotropic static non-SUSY
D2 brane and the change in the angular coordinate converts
spherical 6 to hyperbolic H6. Thus the transformed solution
is,
H (τ )
H˜ (τ )
(dx 1)2
,
β
sin2 θ ,
e2(φ−φ0) = F (τ ) 21
F[
6
] = Qˆ Vol( H6).
The various functions associated with the solution are also
changed under the above rotation and are given below,
F (τ ) =
H (τ )
H˜ (τ )
α
cos2 θ +
H˜ (τ )
H (τ )
ω5
H (τ ) = 1 + τ 5 ,
ω5
H˜ (τ ) = 1 − τ 5 .
Thus we see that the anisotropic static non-SUSY D2 brane
has been converted to anisotropic time-dependent or
spacelike D2 brane. For the former solution the radial coordinate
r was transverse to the D2 brane’s world-volume, whereas,
for the latter the time-like coordinate τ is transverse to the
SD2 brane’s world-volume. The metric of the transverse
sphere d 26 has been converted to negative of the metric of
the hyperbolic space d H 2. The hyperbolic functions sinh2 θ
6
and cosh2 θ become − sin2 θ and cos2 θ respectively,
therefore, the relative sign of the two terms of the function F (τ )
has been flipped. But the form field remains unchanged
with Qˆ → −Qˆ . Thus the first two parameter relations in
(4) remain the same, while the last relation has changed to
Qˆ = 5ω5(α + β) sin 2θ . Now for our purpose we will make
a coordinate transformation from τ to t given by
τ = t
1 + √g(t ) 25
2
,
4ω5
where g(t ) = 1 + t 5
≡ 1 + tt055 .
ds2 = F (τ ) 83 ( H (τ ) H˜ (τ )) 25
4√g(t )
H (τ ) H˜ (τ ) = (1 + √g(t ))2
,
Under this coordinate change we have
ω5 2√g(t )
H (τ ) = 1 + τ 5 = 1 + √g(t )
ω5 2
H˜ (τ ) = 1 − τ 5 = 1 + √g(t )
,
,
(10)
(11)
(12)
Using (10) we can rewrite the anisotropic SD2 brane solution
given in (6) as follows:
It is important to note that in the new coordinate, the original
singularity at τ = ω has been shifted to t = 0. Also note
that as t t0, g(t ), F (t ) → 1 and therefore, the solution
reduces to flat space. In the next two sections we will impose
the assumption t ∼ t0 and also t t0 into the solution
(11) to see how one can get accelerating cosmology in the
first case and a Kasner-like cosmology in the second case in
(3+1) dimensions upon compactification.
3 Compactification and accelerating cosmology
In this section we will compactify the anisotropic SD2
brane solution given in (11) on a six-dimensional hyperbolic
space of a time-dependent volume and write the resultant
four-dimensional metric in the Einstein frame.1 This
fourdimensional metric will have the standard FLRW form whose
cosmology we want to study. We rewrite the metric in (11) in
a four-dimensional part and the transverse six-dimensional
part as
ds2 = ds42 + e2ψ d H62
(13)
1 Here one might ask: since hyperbolic spaces are in general
noncompact in what sense are we compactifying the ten-dimensional space
on six-dimensional hyperbolic space and studying the four-dimensional
cosmology? To address this question we remark that it is quite well
known how to construct compact hyperbolic manifolds (CHMs) from
hyperbolic spaces and there is a vast mathematical literature some of
which are given in [
2–4
]. In short, the CHMs are obtained from Hd
(with d ≥ 2), the universal covering space of d-dimensional
hyperbolic manifold by modding out by an appropriate freely acting discrete
subgroup of the isometry group SO(1, d) of Hd . CHMs have many
interesting properties and we refer the reader to some of the original
literature [
2–4
] for details.
wherδe1 ψ is the radion field and e2ψ = F (t ) 83
g(t ) 8 + δ220 + δ230 + 51 t 2. The four-dimensional metric ds42 is given
as
ds42 = −F (t ) 8 g(t ) δ81 + δ220 + δ230 − 45 dt 2 + F (t )− 8
3 5
The compactified four-dimensional metric (14) when
expressed in an Einstein frame takes the form [
8
],
Note that in the compactified four-dimensional space there
are three fields, namely gμν , φ, ψ . Now we perform another
coordinate transformation,
dη2 = F (t ) 2 g(t )− 51 + δ21 + δ52 + δ53 t 6dt 2 ⇒,
3
η =
F (t ) 4 g(t )− 110 + δ41 + δ120 + δ130 t 3dt,
3
and rewrite the Einstein frame metric ds42E in the standard
flat FLRW form as
ds42E = −dη2 + si2(η)
(dxi )2
3
i=1
with η being the canonical time and the scale factor si (η) ≡
Si (t ). Note that since si (η) are different for each i , the
cosmology here will be anisotropic. Now because of the
complicated relation between t and η let us define (see, for example
[
23,24
])
mi (t ) ≡
ni (t ) ≡
d ln Si (t )
dt
d2 d
dt 2 ln(Si (t )) + dt ln(Si (t ))
d
dt
ln
Si (t )
A(t )
(19)
and with these one can easily see that mi (t ) > 0 implies that
dsi (η)/dη > 0, amounting to expansion of our universe, and
similarly, ni (t ) > 0 implies that d2si (η)/dη2 > 0,
amounting to acceleration of our universe. Therefore, from (19) it is
clear that in the four-dimensional space-time (15) we get an
accelerated expansion in the i th coordinate direction only if
the parameters mi (t ) and ni (t ) are simultaneously positive in
that direction. It can be checked that, for t t0, accelerating
expansion is not possible at all in any direction. However, it
is possible only if t ∼ t0. In this case the first term in the
harmonic function g(t ) given in (9) is of the same order as
the second. The other parameters of the solution, namely,
δ1, δ2 and δ3 cannot be totally arbitrary. From Eq. (5), we
see that the reality of α and β imposes some restriction on
the value of these three anisotropy parameters. Also it can
be checked that by changing the value of θ does not change
the cosmological behavior of the solution very much. Thus
we have chosen some typical values of these parameters (as
given in the figure) and plotted the functions mi (t ) and ni (t )
in Fig. 1, to show that it is indeed possible to have accelerating
expansions in all three directions.
We notice as shown in (a)–(c) in Fig. 1, we always get
expanding universe (given by the solid blue line) in all three
directions, but the expansion is accelerating only for a short
period of time, i.e., the acceleration is transient (given by
the dotted red line). Also note that since mi (t ) and ni (t )
are different for different i , the cosmology is anisotropic,
however, the anisotropy is not too much.
To understand the accelerating expansion, we can write
down the four-dimensional compactified action from the
original ten-dimensional one and obtain the form of the
potential of the dilaton and the radion field [
8
]. The
tendimensional action has the form,
S =
d10x √−g R − 21 (∂φ)2 − 2 ·16! e−φ/2 F[
26
] . (20)
Reducing the action on a six-dimensional hyperbolic space
H6, the four-dimensional action we get2 [
20,23,24,28
]
S4 =
d4x√−g4E R4E − 21 (∂φ)2 − 24(∂ψ)2 − V (φ, ψ)
(21)
2 Here reduction on the hyperbolic space H6 to obtain the
fourdimensional action is done in the sense decribed in footnote 3. This
has also been done in the references [
20,28
].
3
2
1
-1
F (t ) ∼ t − 52α .
In the above we have absorbed t0 in t . But for case (c) F (t )
has an additional cos2 θ factor which can be absorbed in t as
well as in x 1,2,3. Thus in all cases F (t ) has the form as given
in (24). So, in this near region, the space-time metric (15),
the dilaton and the radion fields take the forms,
ds2 = − t 2 27 − 158α − 5δ41 − δ22 − δ23 dt 2 + t 2 23 − 58α + 3δ22 −δ3 (dx 1)2
+ t 2 23 − 58α −δ2+ 3δ23 (dx 2)2 + t 2 23 − 58α − 5δ41 −δ2−δ3 (dx 3)2
e2(φ−φ0) = t 2 − 58α − 5δ41 + 5δ22 + 5δ23 ,
e2ψ = t 2 21 − 3125 α− 156 δ1− δ82 − δ83 .
Now since we are taking t 1 here, we have to be careful
about the validity of the gravity solution. The gravity solution
will be valid as long as the dilaton remains small and the
curvature of the transverse space in string units also remains
small. These two conditions impose certain restrictions on
the parameters of the solution and they are given as
5
t
(25)
(26)
(27)
where
Here Qˆ is the magnetic charge of the D2 brane given in (1).
Note that because of the hyperbolic space compactification
the potential is always positive irrespective of the charge and
therefore there is always a possibility that the system will be
driven to an accelerating phase (see the [
20
]).
4 Compactification and Kasner-like solution
In this section we will show how a four-dimensional
Kasnerlike cosmological solution follows from the anisotropic SD2
brane solution upon the six-dimensional hyperbolic space
compactification discussed in the previous section. The
compactified action expressed in Einstein frame is given in (15).
We take this four-dimensional metric and express it at early
times, t t0. In this case the function g(t ) can be
approximated by
t05 t05
g(t ) = 1 + t 5 ≈ t 5 ∼ t −5.
Also since we want to express the metric components in (15)
as some powers of t , we note from the form of F (t ) in (12)
that this can be done (assuming α > 0 without any loss of
generality) in three ways as follows. (a) Put θ = 0, with
α, β as given in (5), (b) put α = −β = −(3/4)δ1, with θ
arbitrary and (c) both α > 0, β > 0, with θ arbitrary. There
is another possibility with θ = π/2 and β < 0, but this case
can be seen to be equivalent to case (a). Note that for case
(a) and (b) we have Qˆ = 0 (since Qˆ = 5ω5(α + β) sin 2θ ),
however, for case (c) Qˆ is non-zero and the non-SUSY brane
is magnetically charged. In either case (a) or (b) we have
(22)
(23)
(24)
5α + 5δ1 − 4δ2 − 4δ3 > 4
α + 2δ1 − 4δ2 − 4δ3 ≤ 0
where α is as given in (5). Furthermore, the reality of α also
restricts the parameters as
35 2
4 δ1 + 48δ22 + 48δ32 + 16δ2δ3 ≤ 48.
We have checked numerically that all these three conditions
can be satisfied simultaneously for a certain range of values
of the parameters δ1, δ2 and δ3 and only for those values we
have a valid gravity solution (25). We would like to remark
here that the validity of the supergravity solution also requires
that we cannot take t arbitrarily close to zero as we are
considering t 1. In fact t has to be much larger than the
string scale if the supergravity solution remains valid. This
can be seen if we calculate φ˙ 2, ψ˙ 2 and also the scalar
curvature with the solution given in (25). All of these terms come
out to be proportional to 1/t 2 and so, when t 1, they
can become very large invalidating the supergravity solution
and stringy corrections must be included. To avoid this we
require √α t t0 or in terms of scaled t , we must
have √α /t0 t 1. Now, keeping those restrictions in
mind, we can rewrite the solution in terms of canonical time
9 5δ1
8t 2 − 158α − 4 − δ22 − δ23
η ≡ 36−15α−10δ1−4δ2−4δ3 as
ds2 = −dη2 + η2 p1 (dx 1)2 + η2 p2 (dx 2)2 + η2 p3 (dx 3)2,
e2(φ−φ0) = C (δ1, δ2, δ3) η2γφ e2ψ = D(δ1, δ2, δ3)η2γψ .
Note that in writing the metric in (28) we have rescaled the
coordinates x 1, x 2 and x 3 by some constant factors involving
the parameters δ1, δ2, δ3. Also in the dilaton and the radion
field C and D are constants involving these paramaters whose
explicit form will not be important. It can be easily checked
in the defining relation of η that the coefficient in front of t is
always positive definite and that also ensures that as t → 0,
η → 0. The Kasner exponents p1, p2 and p3 in the metric
and γφ , γψ are defined as
1
Rμν,E − 2 ∂μφ∂ν φ − 24∂μψ ∂ν ψ = 0
1
√−gE
1
√−gE
∂μ
∂μ
√−gE gμEν ∂ν φ
√−gE gμEν ∂ν ψ
= 0,
= 0;
here μ, ν run over (1+3)-dimensional space-time. Note that
since we have t 1, the potential in (22) is trivial (the first
term is zero even when Qˆ = 0 because the exponential factor
effectively goes to zero due to the relations given in (26) and
similarly the exponential in the second term also effectively
goes to zero because of (26)). Substituting the above solution
(28) in (30), we get two conditions
2 2 2 1 2
p1 + p2 + p3 = 1, and p1 + p2 + p3 = 1 − 2 γφ − 24γψ2 .
12 − 5α + 12δ2 − 8δ3
p1 = 36 − 15α − 10δ1 − 4δ2 − 4δ3
12 − 5α − 8δ2 + 12δ3
p2 = 36 − 15α − 10δ1 − 4δ2 − 4δ3
12 − 5α − 10δ1 − 8δ2 − 8δ3
p3 = 36 − 15α − 10δ1 − 4δ2 − 4δ3
−5α − 10δ1 + 20δ2 + 20δ3
γφ = 36 − 15α − 10δ1 − 4δ2 − 4δ3
1 16 − 15α − 10δ1 − 4δ2 − 4δ3
γψ = 4 36 − 15α − 10δ1 − 4δ2 − 4δ3
.
Now since this a solution to the compactified
four-dimensional action given in (21), it must satisfy the equations of
motion. The Einstein equation, the dilaton and radion
equations following from (21) have the forms
(28)
(29)
(30)
(31)
The first condition of (31) can be seen to be satisfied trivially
from (29). On the other hand when we substitute the
parameter values from (29) to the second condition of (31), we
find that it gives the same parametric relation as the second
relation of (4) verifying the consistency of the solution. This
therefore shows how one can get a four-dimensional
Kasnerlike solution from the ten-dimensional anisotropic SD2 brane
solution by six-dimensional hyperbolic space
compactification. It is well known that the standard Kasner solution [
25
],
obtained as the solution of the vacuum Einstein equation,
does not lead to expansions in all spatial directions. The
reason is that in standard Kasner cosmology the Kasner
exponents satisfy p1 + p2 + p3 = 1 and p12 + p22 + p32 = 1. Since
these two conditions cannot be satisfied together when the pi
are all positive, the expansions cannot occur in all the
directions. However, for the four-dimensional Kasner cosmology
we obtained from string theory solutions, the parameters pi
can all be positive definite because the second condition here
(31) is different. This is essentially the reason that we can
have expansions in all the directions, but it can easily be
checked that the expansions are decelerating.
5 Conclusion
To summarize, in this paper we have constructed an
anisotropic SD2 brane solution starting from an anisotropic
nonSUSY D2 brane solution of type IIA string theory by the
standard trick of a double Wick rotation. We wanted to see
whether it is possible to generate accelerating cosmologies
in all the directions which is known for the isotropic SD2
brane solution upon compactification on the six-dimensional
hyperbolic space of a time-dependent volume. Indeed we
found that when the resultant four-dimensional metric is
expressed in Einstein frame there are some windows of the
parameters of the solution where one can get accelerating
cosmologies in all the directions and is discussed in Sect. 3.
Here both the expansions and the accelerations we found are
anisotropic. But in order to get accelerating expansions we
noted that the anisotropy cannot be too drastic in three
different directions. We also noted that accelerations are possible
only for t ∼ t0, where t0 is some characteristic time given as
one of the parameters of the solution. Next, we looked at the
four-dimensional metric at early times, i.e., for t t0 and
found that in a suitable coordinate and under certain
conditions on the parameters of the solution, it can be expressed
in a standard four-dimensional Kasner-like form. But unlike
in the standard Kasner cosmology, where expansions in all
three directions are not possible, here we can get expansions
in all the three directions. The reason is that in this case the
relations among the Kasner exponents get modified due to
the presence of the dilaton and the radion field. It would be
interesting to see what effect (such modification to Kasner
solution at early time) it would have on the cosmological
singularities [
29–31
].
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